Modified Young's Double Slit Experiment (YDSE) by Changing Medium

The Young's Double Slit Experiment (YDSE) is a classic demonstration of the wave nature of light, showcasing interference patterns that result when coherent light waves pass through two closely spaced slits and overlap. However, when the medium through which the light travels is changed, the interference pattern is also modified. This modification can be understood and calculated by considering the effects of the new medium on the wavelength and speed of light.

Basic Principle of YDSE

In the standard YDSE, monochromatic light of wavelength $\lambda$ in air (or vacuum) passes through two slits separated by a distance $d$, creating two coherent light sources. These light waves interfere constructively or destructively at different points on a screen placed at a distance $L$ from the slits, forming bright and dark fringes.

The condition for constructive interference (bright fringes) is given by:

$$ d \sin \theta = m\lambda \quad \text{where } m = 0, \pm1, \pm2, \ldots $$

And for destructive interference (dark fringes):

$$ d \sin \theta = \left(m + \frac{1}{2}\right)\lambda \quad \text{where } m = 0, \pm1, \pm2, \ldots $$

Here, $\theta$ is the angle of the fringe with respect to the central axis, and $m$ is the order of the fringe.

Effect of Changing Medium

When the medium is changed, the wavelength of light in the new medium ($\lambda'$) is different from that in air ($\lambda$). The speed of light in the new medium ($v'$) is also different from the speed of light in air ($c$). The relationship between the speed of light, wavelength, and frequency ($f$) is given by:

$$ v = f\lambda $$

Since the frequency of light remains constant when changing the medium, we can write:

$$ \frac{c}{\lambda} = \frac{v'}{\lambda'} $$

Using the refractive index $n$ of the new medium, where $n = \frac{c}{v'}$, we get:

$$ \lambda' = \frac{\lambda}{n} $$

This change in wavelength affects the interference pattern. The fringe width ($\beta$), which is the distance between two consecutive bright or dark fringes, is given by:

$$ \beta = \frac{\lambda L}{d} $$

In the new medium, the modified fringe width ($\beta'$) becomes:

$$ \beta' = \frac{\lambda' L}{d} = \frac{\lambda L}{nd} $$

Table of Differences and Important Points

Property	In Air (Original YDSE)	In New Medium (Modified YDSE)
Wavelength ($\lambda$)	$\lambda$	$\lambda' = \frac{\lambda}{n}$
Speed of Light ($v$)	$c$	$v' = \frac{c}{n}$
Refractive Index ($n$)	1 (for air)	$> 1$ (for denser mediums)
Fringe Width ($\beta$)	$\frac{\lambda L}{d}$	$\frac{\lambda L}{nd}$
Path Difference	$d \sin \theta$	$d \sin \theta$ (unchanged)
Condition for Bright Fringes	$d \sin \theta = m\lambda$	$d \sin \theta = m\lambda'$
Condition for Dark Fringes	$d \sin \theta = \left(m + \frac{1}{2}\right)\lambda$	$d \sin \theta = \left(m + \frac{1}{2}\right)\lambda'$

Examples to Explain Important Points

Example 1: Change in Fringe Width

Suppose we have a YDSE setup with $\lambda = 600 \text{ nm}$ in air, $d = 0.1 \text{ mm}$, and $L = 1 \text{ m}$. The fringe width in air is:

$$ \beta = \frac{\lambda L}{d} = \frac{600 \times 10^{-9} \text{ m} \times 1 \text{ m}}{0.1 \times 10^{-3} \text{ m}} = 6 \text{ mm} $$

If the experiment is now performed in water with a refractive index of $n = 1.33$, the fringe width becomes:

$$ \beta' = \frac{\lambda L}{nd} = \frac{600 \times 10^{-9} \text{ m} \times 1 \text{ m}}{1.33 \times 0.1 \times 10^{-3} \text{ m}} \approx 4.5 \text{ mm} $$

Thus, the fringe width decreases when the medium is changed from air to water.

Example 2: Change in Interference Conditions

If we are looking for the position of the third-order bright fringe ($m = 3$) in air, we use:

$$ d \sin \theta = m\lambda = 3 \times 600 \text{ nm} $$

In water, the condition changes to:

$$ d \sin \theta = m\lambda' = 3 \times \frac{600 \text{ nm}}{1.33} $$

This means that the angle $\theta$ at which this bright fringe appears will be different in water compared to air, leading to a shift in the position of the fringes on the screen.

In conclusion, changing the medium in a YDSE setup significantly affects the interference pattern observed. The wavelength of light in the new medium is reduced by a factor equal to the refractive index, leading to a decrease in fringe width and a shift in the fringe positions. Understanding these changes is crucial for accurately interpreting and applying the results of the modified YDSE in various scientific and engineering contexts.